How to use a sigmoidal function in a multiple (non)linear regression

Question

My data follows a sigmoidal function of the form
$$y=asym/(1+e^{(xmid-x)/scale)})$$
I have taken the function from the SSLogis function in R.
My supervisor and I think that there is a second variable that influences the asymptote and scale of the function.

So I'm stuck with the questions:
1. Is it at all possible to do a multiple regression based on this function?
If yes, 2. Is it possible to transform the sigmoidal to a linear function and use it in a multiple linear regression?
Or 3. Is there a way to do a multiple non-linear regression?

Any help is really welcome.

Answer 1

Certainly you can have multiple predictors in nonlinear least squares.

Exactly how it works depends on how, precisely, you want the second variable to enter.

You could make an equation that (at a fixed asymptote) has a linear predictor by simple reparameterization:

Let $\alpha$ be your "asym" and let $\beta_0=-$xmid/scale. Let $\beta_1$ be 1/scale and let $x_1$ be your $x$.
Let $x_2$ be the new predictor (independent variable, explanatory variable), and let $\beta_2$ be its coefficient.

Then you could write

$$E(y)=\frac{\alpha}{(1+e^{-(\beta_0+\beta_1x_1+\beta_2x_2)})}\quad^*$$

and then $\beta_2=0$ would correspond to your original model.

$^*$ (note that the LHS is $E(y)$ not $y$; the model for $y$ includes an error term)

Whether this is what you need depends on how you see the second variable coming in.